Properties

Label 1728.2.i.n.577.4
Level $1728$
Weight $2$
Character 1728.577
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,2,Mod(577,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.170772624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.4
Root \(1.41203 - 0.0786378i\) of defining polynomial
Character \(\chi\) \(=\) 1728.577
Dual form 1728.2.i.n.1153.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.68614 - 2.92048i) q^{5} +(2.35143 + 4.07279i) q^{7} +O(q^{10})\) \(q+(1.68614 - 2.92048i) q^{5} +(2.35143 + 4.07279i) q^{7} +(0.437696 + 0.758112i) q^{11} +(-0.686141 + 1.18843i) q^{13} +2.37228 q^{17} +5.57825 q^{19} +(-2.35143 + 4.07279i) q^{23} +(-3.18614 - 5.51856i) q^{25} +(2.68614 + 4.65253i) q^{29} +(-3.22682 + 5.58902i) q^{31} +15.8593 q^{35} -4.00000 q^{37} +(0.500000 - 0.866025i) q^{41} +(0.437696 + 0.758112i) q^{43} +(-2.35143 - 4.07279i) q^{47} +(-7.55842 + 13.0916i) q^{49} -4.00000 q^{53} +2.95207 q^{55} +(4.26516 - 7.38747i) q^{59} +(-1.05842 - 1.83324i) q^{61} +(2.31386 + 4.00772i) q^{65} +(4.26516 - 7.38747i) q^{67} +9.40571 q^{71} +10.3723 q^{73} +(-2.05842 + 3.56529i) q^{77} +(3.22682 + 5.58902i) q^{79} +(-1.47603 - 2.55657i) q^{83} +(4.00000 - 6.92820i) q^{85} -12.7446 q^{89} -6.45364 q^{91} +(9.40571 - 16.2912i) q^{95} +(-4.50000 - 7.79423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} + 6 q^{13} - 4 q^{17} - 14 q^{25} + 10 q^{29} - 32 q^{37} + 4 q^{41} - 26 q^{49} - 32 q^{53} + 26 q^{61} + 30 q^{65} + 60 q^{73} + 18 q^{77} + 32 q^{85} - 56 q^{89} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.68614 2.92048i 0.754065 1.30608i −0.191773 0.981439i \(-0.561424\pi\)
0.945838 0.324640i \(-0.105243\pi\)
\(6\) 0 0
\(7\) 2.35143 + 4.07279i 0.888756 + 1.53937i 0.841347 + 0.540495i \(0.181763\pi\)
0.0474088 + 0.998876i \(0.484904\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.437696 + 0.758112i 0.131970 + 0.228579i 0.924436 0.381337i \(-0.124536\pi\)
−0.792466 + 0.609917i \(0.791203\pi\)
\(12\) 0 0
\(13\) −0.686141 + 1.18843i −0.190301 + 0.329611i −0.945350 0.326057i \(-0.894280\pi\)
0.755049 + 0.655669i \(0.227613\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.37228 0.575363 0.287681 0.957726i \(-0.407116\pi\)
0.287681 + 0.957726i \(0.407116\pi\)
\(18\) 0 0
\(19\) 5.57825 1.27974 0.639869 0.768484i \(-0.278989\pi\)
0.639869 + 0.768484i \(0.278989\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.35143 + 4.07279i −0.490307 + 0.849236i −0.999938 0.0111571i \(-0.996448\pi\)
0.509631 + 0.860393i \(0.329782\pi\)
\(24\) 0 0
\(25\) −3.18614 5.51856i −0.637228 1.10371i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.68614 + 4.65253i 0.498804 + 0.863954i 0.999999 0.00138070i \(-0.000439492\pi\)
−0.501195 + 0.865334i \(0.667106\pi\)
\(30\) 0 0
\(31\) −3.22682 + 5.58902i −0.579554 + 1.00382i 0.415976 + 0.909375i \(0.363440\pi\)
−0.995530 + 0.0944415i \(0.969893\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 15.8593 2.68072
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.500000 0.866025i 0.0780869 0.135250i −0.824338 0.566099i \(-0.808452\pi\)
0.902424 + 0.430848i \(0.141786\pi\)
\(42\) 0 0
\(43\) 0.437696 + 0.758112i 0.0667481 + 0.115611i 0.897468 0.441079i \(-0.145404\pi\)
−0.830720 + 0.556690i \(0.812071\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.35143 4.07279i −0.342991 0.594078i 0.641996 0.766708i \(-0.278107\pi\)
−0.984987 + 0.172630i \(0.944773\pi\)
\(48\) 0 0
\(49\) −7.55842 + 13.0916i −1.07977 + 1.87022i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 2.95207 0.398057
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.26516 7.38747i 0.555276 0.961767i −0.442606 0.896716i \(-0.645946\pi\)
0.997882 0.0650505i \(-0.0207208\pi\)
\(60\) 0 0
\(61\) −1.05842 1.83324i −0.135517 0.234722i 0.790278 0.612749i \(-0.209936\pi\)
−0.925795 + 0.378026i \(0.876603\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.31386 + 4.00772i 0.286999 + 0.497097i
\(66\) 0 0
\(67\) 4.26516 7.38747i 0.521072 0.902523i −0.478628 0.878018i \(-0.658866\pi\)
0.999700 0.0245053i \(-0.00780106\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.40571 1.11625 0.558126 0.829756i \(-0.311520\pi\)
0.558126 + 0.829756i \(0.311520\pi\)
\(72\) 0 0
\(73\) 10.3723 1.21398 0.606992 0.794708i \(-0.292376\pi\)
0.606992 + 0.794708i \(0.292376\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.05842 + 3.56529i −0.234579 + 0.406303i
\(78\) 0 0
\(79\) 3.22682 + 5.58902i 0.363046 + 0.628813i 0.988460 0.151479i \(-0.0484036\pi\)
−0.625415 + 0.780292i \(0.715070\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.47603 2.55657i −0.162016 0.280620i 0.773576 0.633704i \(-0.218466\pi\)
−0.935592 + 0.353084i \(0.885133\pi\)
\(84\) 0 0
\(85\) 4.00000 6.92820i 0.433861 0.751469i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.7446 −1.35092 −0.675460 0.737396i \(-0.736055\pi\)
−0.675460 + 0.737396i \(0.736055\pi\)
\(90\) 0 0
\(91\) −6.45364 −0.676525
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.40571 16.2912i 0.965005 1.67144i
\(96\) 0 0
\(97\) −4.50000 7.79423i −0.456906 0.791384i 0.541890 0.840450i \(-0.317709\pi\)
−0.998796 + 0.0490655i \(0.984376\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.05842 1.83324i −0.105317 0.182414i 0.808551 0.588426i \(-0.200252\pi\)
−0.913868 + 0.406012i \(0.866919\pi\)
\(102\) 0 0
\(103\) −2.35143 + 4.07279i −0.231693 + 0.401304i −0.958306 0.285742i \(-0.907760\pi\)
0.726613 + 0.687047i \(0.241093\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.57825 −0.539270 −0.269635 0.962963i \(-0.586903\pi\)
−0.269635 + 0.962963i \(0.586903\pi\)
\(108\) 0 0
\(109\) 5.48913 0.525763 0.262881 0.964828i \(-0.415327\pi\)
0.262881 + 0.964828i \(0.415327\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.68614 15.0448i 0.817123 1.41530i −0.0906698 0.995881i \(-0.528901\pi\)
0.907793 0.419418i \(-0.137766\pi\)
\(114\) 0 0
\(115\) 7.92967 + 13.7346i 0.739446 + 1.28076i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.57825 + 9.66181i 0.511357 + 0.885696i
\(120\) 0 0
\(121\) 5.11684 8.86263i 0.465168 0.805694i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.62772 −0.413916
\(126\) 0 0
\(127\) 11.1565 0.989979 0.494989 0.868899i \(-0.335172\pi\)
0.494989 + 0.868899i \(0.335172\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.22682 5.58902i 0.281929 0.488315i −0.689931 0.723875i \(-0.742359\pi\)
0.971860 + 0.235560i \(0.0756925\pi\)
\(132\) 0 0
\(133\) 13.1168 + 22.7190i 1.13737 + 1.96999i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.24456 + 9.08385i 0.448073 + 0.776086i 0.998261 0.0589556i \(-0.0187770\pi\)
−0.550187 + 0.835041i \(0.685444\pi\)
\(138\) 0 0
\(139\) −9.84341 + 17.0493i −0.834907 + 1.44610i 0.0591995 + 0.998246i \(0.481145\pi\)
−0.894106 + 0.447855i \(0.852188\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.20128 −0.100456
\(144\) 0 0
\(145\) 18.1168 1.50452
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.68614 15.0448i 0.711596 1.23252i −0.252661 0.967555i \(-0.581306\pi\)
0.964258 0.264966i \(-0.0853608\pi\)
\(150\) 0 0
\(151\) −2.35143 4.07279i −0.191356 0.331439i 0.754344 0.656480i \(-0.227955\pi\)
−0.945700 + 0.325041i \(0.894622\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 10.8817 + 18.8477i 0.874043 + 1.51389i
\(156\) 0 0
\(157\) 7.05842 12.2255i 0.563323 0.975705i −0.433880 0.900971i \(-0.642856\pi\)
0.997203 0.0747341i \(-0.0238108\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −22.1168 −1.74305
\(162\) 0 0
\(163\) 18.8114 1.47342 0.736712 0.676207i \(-0.236377\pi\)
0.736712 + 0.676207i \(0.236377\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.22682 + 5.58902i −0.249699 + 0.432491i −0.963442 0.267916i \(-0.913665\pi\)
0.713743 + 0.700407i \(0.246998\pi\)
\(168\) 0 0
\(169\) 5.55842 + 9.62747i 0.427571 + 0.740575i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.68614 4.65253i −0.204223 0.353725i 0.745662 0.666325i \(-0.232134\pi\)
−0.949885 + 0.312599i \(0.898800\pi\)
\(174\) 0 0
\(175\) 14.9840 25.9530i 1.13268 1.96186i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −18.8114 −1.40603 −0.703016 0.711174i \(-0.748164\pi\)
−0.703016 + 0.711174i \(0.748164\pi\)
\(180\) 0 0
\(181\) −26.2337 −1.94993 −0.974967 0.222348i \(-0.928628\pi\)
−0.974967 + 0.222348i \(0.928628\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.74456 + 11.6819i −0.495870 + 0.858872i
\(186\) 0 0
\(187\) 1.03834 + 1.79846i 0.0759308 + 0.131516i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.80507 + 15.2508i 0.637112 + 1.10351i 0.986063 + 0.166370i \(0.0532046\pi\)
−0.348951 + 0.937141i \(0.613462\pi\)
\(192\) 0 0
\(193\) −0.500000 + 0.866025i −0.0359908 + 0.0623379i −0.883460 0.468507i \(-0.844792\pi\)
0.847469 + 0.530845i \(0.178125\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.7446 −0.765518 −0.382759 0.923848i \(-0.625026\pi\)
−0.382759 + 0.923848i \(0.625026\pi\)
\(198\) 0 0
\(199\) −17.0606 −1.20940 −0.604698 0.796455i \(-0.706706\pi\)
−0.604698 + 0.796455i \(0.706706\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.6325 + 21.8802i −0.886630 + 1.53569i
\(204\) 0 0
\(205\) −1.68614 2.92048i −0.117765 0.203975i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.44158 + 4.22894i 0.168887 + 0.292522i
\(210\) 0 0
\(211\) −7.92967 + 13.7346i −0.545901 + 0.945529i 0.452648 + 0.891689i \(0.350479\pi\)
−0.998550 + 0.0538397i \(0.982854\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.95207 0.201329
\(216\) 0 0
\(217\) −30.3505 −2.06033
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.62772 + 2.81929i −0.109492 + 0.189646i
\(222\) 0 0
\(223\) 2.35143 + 4.07279i 0.157463 + 0.272734i 0.933953 0.357395i \(-0.116335\pi\)
−0.776490 + 0.630130i \(0.783002\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.7188 18.5655i −0.711432 1.23224i −0.964320 0.264740i \(-0.914714\pi\)
0.252888 0.967496i \(-0.418620\pi\)
\(228\) 0 0
\(229\) 0.686141 1.18843i 0.0453415 0.0785337i −0.842464 0.538753i \(-0.818896\pi\)
0.887805 + 0.460219i \(0.152229\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 25.8614 1.69424 0.847119 0.531404i \(-0.178335\pi\)
0.847119 + 0.531404i \(0.178335\pi\)
\(234\) 0 0
\(235\) −15.8593 −1.03455
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.80507 15.2508i 0.569552 0.986494i −0.427058 0.904224i \(-0.640450\pi\)
0.996610 0.0822694i \(-0.0262168\pi\)
\(240\) 0 0
\(241\) −5.87228 10.1711i −0.378267 0.655177i 0.612543 0.790437i \(-0.290147\pi\)
−0.990810 + 0.135260i \(0.956813\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 25.4891 + 44.1485i 1.62844 + 2.82054i
\(246\) 0 0
\(247\) −3.82746 + 6.62936i −0.243536 + 0.421816i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −16.7347 −1.05629 −0.528144 0.849155i \(-0.677112\pi\)
−0.528144 + 0.849155i \(0.677112\pi\)
\(252\) 0 0
\(253\) −4.11684 −0.258824
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.24456 14.2800i 0.514282 0.890762i −0.485581 0.874192i \(-0.661392\pi\)
0.999863 0.0165703i \(-0.00527474\pi\)
\(258\) 0 0
\(259\) −9.40571 16.2912i −0.584442 1.01228i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −8.80507 15.2508i −0.542944 0.940406i −0.998733 0.0503185i \(-0.983976\pi\)
0.455790 0.890088i \(-0.349357\pi\)
\(264\) 0 0
\(265\) −6.74456 + 11.6819i −0.414315 + 0.717615i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −13.4891 −0.822446 −0.411223 0.911535i \(-0.634898\pi\)
−0.411223 + 0.911535i \(0.634898\pi\)
\(270\) 0 0
\(271\) 22.3130 1.35542 0.677709 0.735330i \(-0.262973\pi\)
0.677709 + 0.735330i \(0.262973\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.78912 4.83090i 0.168190 0.291314i
\(276\) 0 0
\(277\) −2.05842 3.56529i −0.123679 0.214218i 0.797537 0.603270i \(-0.206136\pi\)
−0.921216 + 0.389052i \(0.872803\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.43070 2.47805i −0.0853486 0.147828i 0.820191 0.572090i \(-0.193867\pi\)
−0.905540 + 0.424262i \(0.860534\pi\)
\(282\) 0 0
\(283\) 7.92967 13.7346i 0.471370 0.816437i −0.528093 0.849186i \(-0.677093\pi\)
0.999464 + 0.0327491i \(0.0104262\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.70285 0.277601
\(288\) 0 0
\(289\) −11.3723 −0.668958
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.31386 12.6680i 0.427280 0.740071i −0.569350 0.822095i \(-0.692805\pi\)
0.996630 + 0.0820241i \(0.0261384\pi\)
\(294\) 0 0
\(295\) −14.3833 24.9126i −0.837429 1.45047i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.22682 5.58902i −0.186612 0.323221i
\(300\) 0 0
\(301\) −2.05842 + 3.56529i −0.118645 + 0.205500i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.13859 −0.408755
\(306\) 0 0
\(307\) −20.8881 −1.19215 −0.596073 0.802930i \(-0.703273\pi\)
−0.596073 + 0.802930i \(0.703273\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −13.5079 + 23.3964i −0.765964 + 1.32669i 0.173771 + 0.984786i \(0.444405\pi\)
−0.939735 + 0.341903i \(0.888929\pi\)
\(312\) 0 0
\(313\) 7.61684 + 13.1928i 0.430529 + 0.745699i 0.996919 0.0784388i \(-0.0249935\pi\)
−0.566389 + 0.824138i \(0.691660\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.31386 2.27567i −0.0737937 0.127814i 0.826767 0.562544i \(-0.190177\pi\)
−0.900561 + 0.434730i \(0.856844\pi\)
\(318\) 0 0
\(319\) −2.35143 + 4.07279i −0.131655 + 0.228033i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 13.2332 0.736313
\(324\) 0 0
\(325\) 8.74456 0.485061
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.0584 19.1537i 0.609671 1.05598i
\(330\) 0 0
\(331\) −4.97760 8.62146i −0.273594 0.473879i 0.696186 0.717862i \(-0.254879\pi\)
−0.969779 + 0.243983i \(0.921546\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −14.3833 24.9126i −0.785844 1.36112i
\(336\) 0 0
\(337\) 4.50000 7.79423i 0.245131 0.424579i −0.717038 0.697034i \(-0.754502\pi\)
0.962168 + 0.272456i \(0.0878358\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5.64947 −0.305936
\(342\) 0 0
\(343\) −38.1723 −2.06111
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.26516 + 7.38747i −0.228966 + 0.396580i −0.957502 0.288427i \(-0.906868\pi\)
0.728536 + 0.685007i \(0.240201\pi\)
\(348\) 0 0
\(349\) −2.94158 5.09496i −0.157459 0.272727i 0.776493 0.630126i \(-0.216997\pi\)
−0.933952 + 0.357399i \(0.883664\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.61684 + 13.1928i 0.405404 + 0.702180i 0.994368 0.105979i \(-0.0337976\pi\)
−0.588965 + 0.808159i \(0.700464\pi\)
\(354\) 0 0
\(355\) 15.8593 27.4692i 0.841727 1.45791i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −33.4695 −1.76645 −0.883226 0.468948i \(-0.844633\pi\)
−0.883226 + 0.468948i \(0.844633\pi\)
\(360\) 0 0
\(361\) 12.1168 0.637729
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 17.4891 30.2921i 0.915423 1.58556i
\(366\) 0 0
\(367\) −15.5846 26.9933i −0.813509 1.40904i −0.910393 0.413744i \(-0.864221\pi\)
0.0968838 0.995296i \(-0.469112\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −9.40571 16.2912i −0.488320 0.845795i
\(372\) 0 0
\(373\) 0.0584220 0.101190i 0.00302498 0.00523941i −0.864509 0.502617i \(-0.832370\pi\)
0.867534 + 0.497378i \(0.165704\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.37228 −0.379692
\(378\) 0 0
\(379\) −3.82746 −0.196604 −0.0983018 0.995157i \(-0.531341\pi\)
−0.0983018 + 0.995157i \(0.531341\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.8817 + 18.8477i −0.556031 + 0.963074i 0.441791 + 0.897118i \(0.354343\pi\)
−0.997823 + 0.0659564i \(0.978990\pi\)
\(384\) 0 0
\(385\) 6.94158 + 12.0232i 0.353776 + 0.612757i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −16.1753 28.0164i −0.820119 1.42049i −0.905594 0.424147i \(-0.860574\pi\)
0.0854750 0.996340i \(-0.472759\pi\)
\(390\) 0 0
\(391\) −5.57825 + 9.66181i −0.282104 + 0.488619i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 21.7635 1.09504
\(396\) 0 0
\(397\) 4.00000 0.200754 0.100377 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.98913 + 15.5696i −0.448895 + 0.777510i −0.998314 0.0580372i \(-0.981516\pi\)
0.549419 + 0.835547i \(0.314849\pi\)
\(402\) 0 0
\(403\) −4.42810 7.66970i −0.220580 0.382055i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.75079 3.03245i −0.0867832 0.150313i
\(408\) 0 0
\(409\) 14.8723 25.7595i 0.735387 1.27373i −0.219166 0.975688i \(-0.570334\pi\)
0.954553 0.298040i \(-0.0963329\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 40.1168 1.97402
\(414\) 0 0
\(415\) −9.95521 −0.488682
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.92967 13.7346i 0.387390 0.670979i −0.604708 0.796448i \(-0.706710\pi\)
0.992098 + 0.125468i \(0.0400434\pi\)
\(420\) 0 0
\(421\) −8.31386 14.4000i −0.405193 0.701814i 0.589151 0.808023i \(-0.299462\pi\)
−0.994344 + 0.106208i \(0.966129\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.55842 13.0916i −0.366637 0.635034i
\(426\) 0 0
\(427\) 4.97760 8.62146i 0.240883 0.417222i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −11.1565 −0.537389 −0.268695 0.963225i \(-0.586592\pi\)
−0.268695 + 0.963225i \(0.586592\pi\)
\(432\) 0 0
\(433\) −0.883156 −0.0424418 −0.0212209 0.999775i \(-0.506755\pi\)
−0.0212209 + 0.999775i \(0.506755\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −13.1168 + 22.7190i −0.627464 + 1.08680i
\(438\) 0 0
\(439\) 10.8817 + 18.8477i 0.519357 + 0.899553i 0.999747 + 0.0224981i \(0.00716196\pi\)
−0.480390 + 0.877055i \(0.659505\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.7188 + 18.5655i 0.509265 + 0.882074i 0.999942 + 0.0107321i \(0.00341620\pi\)
−0.490677 + 0.871342i \(0.663250\pi\)
\(444\) 0 0
\(445\) −21.4891 + 37.2203i −1.01868 + 1.76441i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.883156 0.0416787 0.0208394 0.999783i \(-0.493366\pi\)
0.0208394 + 0.999783i \(0.493366\pi\)
\(450\) 0 0
\(451\) 0.875393 0.0412206
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10.8817 + 18.8477i −0.510144 + 0.883595i
\(456\) 0 0
\(457\) −9.98913 17.3017i −0.467272 0.809338i 0.532029 0.846726i \(-0.321430\pi\)
−0.999301 + 0.0373879i \(0.988096\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.94158 + 10.2911i 0.276727 + 0.479305i 0.970569 0.240822i \(-0.0774169\pi\)
−0.693842 + 0.720127i \(0.744084\pi\)
\(462\) 0 0
\(463\) 0.274750 0.475881i 0.0127687 0.0221161i −0.859570 0.511017i \(-0.829269\pi\)
0.872339 + 0.488901i \(0.162602\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9.73160 −0.450325 −0.225162 0.974321i \(-0.572291\pi\)
−0.225162 + 0.974321i \(0.572291\pi\)
\(468\) 0 0
\(469\) 40.1168 1.85242
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.383156 + 0.663646i −0.0176175 + 0.0305145i
\(474\) 0 0
\(475\) −17.7731 30.7839i −0.815485 1.41246i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.35143 + 4.07279i 0.107439 + 0.186091i 0.914732 0.404061i \(-0.132401\pi\)
−0.807293 + 0.590151i \(0.799068\pi\)
\(480\) 0 0
\(481\) 2.74456 4.75372i 0.125141 0.216751i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −30.3505 −1.37815
\(486\) 0 0
\(487\) 11.1565 0.505549 0.252775 0.967525i \(-0.418657\pi\)
0.252775 + 0.967525i \(0.418657\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −20.1245 + 34.8567i −0.908206 + 1.57306i −0.0916519 + 0.995791i \(0.529215\pi\)
−0.816554 + 0.577268i \(0.804119\pi\)
\(492\) 0 0
\(493\) 6.37228 + 11.0371i 0.286993 + 0.497087i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22.1168 + 38.3075i 0.992076 + 1.71833i
\(498\) 0 0
\(499\) 6.89134 11.9361i 0.308499 0.534335i −0.669536 0.742780i \(-0.733507\pi\)
0.978034 + 0.208445i \(0.0668402\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −31.7187 −1.41427 −0.707133 0.707080i \(-0.750012\pi\)
−0.707133 + 0.707080i \(0.750012\pi\)
\(504\) 0 0
\(505\) −7.13859 −0.317663
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −19.1753 + 33.2125i −0.849929 + 1.47212i 0.0313424 + 0.999509i \(0.490022\pi\)
−0.881271 + 0.472611i \(0.843312\pi\)
\(510\) 0 0
\(511\) 24.3897 + 42.2441i 1.07894 + 1.86877i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.92967 + 13.7346i 0.349423 + 0.605219i
\(516\) 0 0
\(517\) 2.05842 3.56529i 0.0905293 0.156801i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −35.3505 −1.54873 −0.774367 0.632736i \(-0.781932\pi\)
−0.774367 + 0.632736i \(0.781932\pi\)
\(522\) 0 0
\(523\) 12.9073 0.564396 0.282198 0.959356i \(-0.408936\pi\)
0.282198 + 0.959356i \(0.408936\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.65492 + 13.2587i −0.333454 + 0.577559i
\(528\) 0 0
\(529\) 0.441578 + 0.764836i 0.0191990 + 0.0332537i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.686141 + 1.18843i 0.0297201 + 0.0514766i
\(534\) 0 0
\(535\) −9.40571 + 16.2912i −0.406644 + 0.704329i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −13.2332 −0.569993
\(540\) 0 0
\(541\) 2.74456 0.117998 0.0589990 0.998258i \(-0.481209\pi\)
0.0589990 + 0.998258i \(0.481209\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 9.25544 16.0309i 0.396459 0.686688i
\(546\) 0 0
\(547\) 5.14055 + 8.90370i 0.219794 + 0.380695i 0.954745 0.297426i \(-0.0961281\pi\)
−0.734951 + 0.678120i \(0.762795\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 14.9840 + 25.9530i 0.638338 + 1.10563i
\(552\) 0 0
\(553\) −15.1753 + 26.2843i −0.645318 + 1.11772i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.25544 −0.307423 −0.153711 0.988116i \(-0.549123\pi\)
−0.153711 + 0.988116i \(0.549123\pi\)
\(558\) 0 0
\(559\) −1.20128 −0.0508089
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.38977 5.87125i 0.142862 0.247444i −0.785711 0.618593i \(-0.787703\pi\)
0.928573 + 0.371150i \(0.121036\pi\)
\(564\) 0 0
\(565\) −29.2921 50.7354i −1.23233 2.13446i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14.1277 24.4699i −0.592265 1.02583i −0.993927 0.110045i \(-0.964901\pi\)
0.401662 0.915788i \(-0.368433\pi\)
\(570\) 0 0
\(571\) −20.1245 + 34.8567i −0.842184 + 1.45871i 0.0458596 + 0.998948i \(0.485397\pi\)
−0.888044 + 0.459758i \(0.847936\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 29.9679 1.24975
\(576\) 0 0
\(577\) 0.883156 0.0367663 0.0183831 0.999831i \(-0.494148\pi\)
0.0183831 + 0.999831i \(0.494148\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.94158 12.0232i 0.287985 0.498805i
\(582\) 0 0
\(583\) −1.75079 3.03245i −0.0725101 0.125591i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.2491 33.3404i −0.794496 1.37611i −0.923159 0.384419i \(-0.874402\pi\)
0.128663 0.991688i \(-0.458932\pi\)
\(588\) 0 0
\(589\) −18.0000 + 31.1769i −0.741677 + 1.28462i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −39.7228 −1.63122 −0.815610 0.578602i \(-0.803599\pi\)
−0.815610 + 0.578602i \(0.803599\pi\)
\(594\) 0 0
\(595\) 37.6228 1.54239
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.5846 26.9933i 0.636769 1.10292i −0.349368 0.936986i \(-0.613604\pi\)
0.986137 0.165931i \(-0.0530630\pi\)
\(600\) 0 0
\(601\) 18.6168 + 32.2453i 0.759397 + 1.31531i 0.943159 + 0.332343i \(0.107839\pi\)
−0.183762 + 0.982971i \(0.558827\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −17.2554 29.8873i −0.701533 1.21509i
\(606\) 0 0
\(607\) −0.274750 + 0.475881i −0.0111518 + 0.0193154i −0.871547 0.490311i \(-0.836883\pi\)
0.860396 + 0.509627i \(0.170216\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.45364 0.261086
\(612\) 0 0
\(613\) 32.4674 1.31134 0.655672 0.755045i \(-0.272385\pi\)
0.655672 + 0.755045i \(0.272385\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.1277 + 22.7379i −0.528502 + 0.915392i 0.470946 + 0.882162i \(0.343913\pi\)
−0.999448 + 0.0332302i \(0.989421\pi\)
\(618\) 0 0
\(619\) 5.14055 + 8.90370i 0.206616 + 0.357870i 0.950646 0.310276i \(-0.100422\pi\)
−0.744030 + 0.668146i \(0.767088\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −29.9679 51.9060i −1.20064 2.07957i
\(624\) 0 0
\(625\) 8.12772 14.0776i 0.325109 0.563105i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.48913 −0.378356
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 18.8114 32.5823i 0.746508 1.29299i
\(636\) 0 0
\(637\) −10.3723 17.9653i −0.410965 0.711812i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.12772 + 1.95327i 0.0445422 + 0.0771494i 0.887437 0.460929i \(-0.152484\pi\)
−0.842895 + 0.538078i \(0.819150\pi\)
\(642\) 0 0
\(643\) −1.31309 + 2.27434i −0.0517832 + 0.0896911i −0.890755 0.454484i \(-0.849824\pi\)
0.838972 + 0.544175i \(0.183157\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.15335 −0.163285 −0.0816426 0.996662i \(-0.526017\pi\)
−0.0816426 + 0.996662i \(0.526017\pi\)
\(648\) 0 0
\(649\) 7.46738 0.293120
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.9416 + 20.6834i −0.467310 + 0.809405i −0.999302 0.0373444i \(-0.988110\pi\)
0.531992 + 0.846749i \(0.321443\pi\)
\(654\) 0 0
\(655\) −10.8817 18.8477i −0.425185 0.736442i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.2875 + 35.1389i 0.790287 + 1.36882i 0.925789 + 0.378040i \(0.123402\pi\)
−0.135502 + 0.990777i \(0.543265\pi\)
\(660\) 0 0
\(661\) 7.05842 12.2255i 0.274541 0.475519i −0.695478 0.718547i \(-0.744807\pi\)
0.970019 + 0.243028i \(0.0781408\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 88.4674 3.43062
\(666\) 0 0
\(667\) −25.2651 −0.978267
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.926535 1.60481i 0.0357685 0.0619528i
\(672\) 0 0
\(673\) 14.1753 + 24.5523i 0.546416 + 0.946421i 0.998516 + 0.0544536i \(0.0173417\pi\)
−0.452100 + 0.891967i \(0.649325\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 25.1753 + 43.6048i 0.967564 + 1.67587i 0.702562 + 0.711622i \(0.252039\pi\)
0.265002 + 0.964248i \(0.414627\pi\)
\(678\) 0 0
\(679\) 21.1628 36.6551i 0.812156 1.40669i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.73160 0.372369 0.186185 0.982515i \(-0.440388\pi\)
0.186185 + 0.982515i \(0.440388\pi\)
\(684\) 0 0
\(685\) 35.3723 1.35151
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.74456 4.75372i 0.104560 0.181102i
\(690\) 0 0
\(691\) 9.13096 + 15.8153i 0.347358 + 0.601642i 0.985779 0.168045i \(-0.0537455\pi\)
−0.638421 + 0.769687i \(0.720412\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.1947 + 57.4950i 1.25915 + 2.18091i
\(696\) 0 0
\(697\) 1.18614 2.05446i 0.0449283 0.0778181i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23.2554 0.878346 0.439173 0.898403i \(-0.355272\pi\)
0.439173 + 0.898403i \(0.355272\pi\)
\(702\) 0 0
\(703\) −22.3130 −0.841550
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.97760 8.62146i 0.187202 0.324244i
\(708\) 0 0
\(709\) 13.8030 + 23.9075i 0.518382 + 0.897864i 0.999772 + 0.0213574i \(0.00679878\pi\)
−0.481390 + 0.876507i \(0.659868\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.1753 26.2843i −0.568318 0.984356i
\(714\) 0 0
\(715\) −2.02554 + 3.50833i −0.0757507 + 0.131204i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15.3098 −0.570961 −0.285481 0.958385i \(-0.592153\pi\)
−0.285481 + 0.958385i \(0.592153\pi\)
\(720\) 0 0
\(721\) −22.1168 −0.823674
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 17.1168 29.6472i 0.635704 1.10107i
\(726\) 0 0
\(727\) −10.0064 17.3315i −0.371115 0.642790i 0.618622 0.785689i \(-0.287691\pi\)
−0.989737 + 0.142898i \(0.954358\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.03834 + 1.79846i 0.0384043 + 0.0665183i
\(732\) 0 0
\(733\) 20.0584 34.7422i 0.740875 1.28323i −0.211223 0.977438i \(-0.567745\pi\)
0.952097 0.305795i \(-0.0989221\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.46738 0.275064
\(738\) 0 0
\(739\) −7.32903 −0.269603 −0.134801 0.990873i \(-0.543040\pi\)
−0.134801 + 0.990873i \(0.543040\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.6644 42.7200i 0.904850 1.56725i 0.0837309 0.996488i \(-0.473316\pi\)
0.821119 0.570757i \(-0.193350\pi\)
\(744\) 0 0
\(745\) −29.2921 50.7354i −1.07318 1.85880i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −13.1168 22.7190i −0.479279 0.830136i
\(750\) 0 0
\(751\) 10.8817 18.8477i 0.397081 0.687764i −0.596284 0.802774i \(-0.703357\pi\)
0.993364 + 0.115010i \(0.0366900\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −15.8593 −0.577181
\(756\) 0 0
\(757\) −34.4674 −1.25274 −0.626369 0.779527i \(-0.715460\pi\)
−0.626369 + 0.779527i \(0.715460\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.31386 12.6680i 0.265127 0.459214i −0.702470 0.711714i \(-0.747919\pi\)
0.967597 + 0.252500i \(0.0812527\pi\)
\(762\) 0 0
\(763\) 12.9073 + 22.3561i 0.467275 + 0.809344i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.85300 + 10.1377i 0.211339 + 0.366051i
\(768\) 0 0
\(769\) 16.0584 27.8140i 0.579082 1.00300i −0.416503 0.909134i \(-0.636745\pi\)
0.995585 0.0938645i \(-0.0299220\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7.25544 −0.260960 −0.130480 0.991451i \(-0.541652\pi\)
−0.130480 + 0.991451i \(0.541652\pi\)
\(774\) 0 0
\(775\) 41.1244 1.47723
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.78912 4.83090i 0.0999307 0.173085i
\(780\) 0 0
\(781\) 4.11684 + 7.13058i 0.147312 + 0.255152i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −23.8030 41.2280i −0.849565 1.47149i
\(786\) 0 0
\(787\) 4.42810 7.66970i 0.157845 0.273395i −0.776246 0.630430i \(-0.782879\pi\)
0.934091 + 0.357034i \(0.116212\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 81.6993 2.90489
\(792\) 0 0
\(793\) 2.90491 0.103156
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.54755 14.8048i 0.302770 0.524412i −0.673993 0.738738i \(-0.735422\pi\)
0.976762 + 0.214326i \(0.0687554\pi\)
\(798\) 0 0
\(799\) −5.57825 9.66181i −0.197344 0.341810i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.53991 + 7.86335i 0.160210 + 0.277492i
\(804\) 0 0
\(805\) −37.2921 + 64.5918i −1.31437 + 2.27656i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.3723 0.645935 0.322968 0.946410i \(-0.395320\pi\)
0.322968 + 0.946410i \(0.395320\pi\)
\(810\) 0 0
\(811\) −50.2042 −1.76291 −0.881454 0.472269i \(-0.843435\pi\)
−0.881454 + 0.472269i \(0.843435\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 31.7187 54.9384i 1.11106 1.92441i
\(816\) 0 0
\(817\) 2.44158 + 4.22894i 0.0854200 + 0.147952i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.31386 2.27567i −0.0458540 0.0794215i 0.842187 0.539185i \(-0.181268\pi\)
−0.888041 + 0.459763i \(0.847934\pi\)
\(822\) 0 0
\(823\) −10.8817 + 18.8477i −0.379314 + 0.656991i −0.990963 0.134139i \(-0.957173\pi\)
0.611649 + 0.791129i \(0.290507\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18.8114 −0.654137 −0.327069 0.945001i \(-0.606061\pi\)
−0.327069 + 0.945001i \(0.606061\pi\)
\(828\) 0 0
\(829\) −26.2337 −0.911134 −0.455567 0.890202i \(-0.650563\pi\)
−0.455567 + 0.890202i \(0.650563\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −17.9307 + 31.0569i −0.621262 + 1.07606i
\(834\) 0 0
\(835\) 10.8817 + 18.8477i 0.376578 + 0.652253i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3.22682 + 5.58902i 0.111402 + 0.192954i 0.916336 0.400411i \(-0.131132\pi\)
−0.804934 + 0.593365i \(0.797799\pi\)
\(840\) 0 0
\(841\) 0.0692967 0.120025i 0.00238954 0.00413881i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 37.4891 1.28967
\(846\) 0 0
\(847\) 48.1275 1.65368
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.40571 16.2912i 0.322424 0.558454i
\(852\) 0 0
\(853\) −1.94158 3.36291i −0.0664784 0.115144i 0.830870 0.556466i \(-0.187843\pi\)
−0.897349 + 0.441322i \(0.854510\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.54755 + 16.5368i 0.326138 + 0.564888i 0.981742 0.190217i \(-0.0609193\pi\)
−0.655604 + 0.755105i \(0.727586\pi\)
\(858\) 0 0
\(859\) −1.31309 + 2.27434i −0.0448020 + 0.0775994i −0.887557 0.460698i \(-0.847599\pi\)
0.842755 + 0.538298i \(0.180932\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 37.6228 1.28070 0.640348 0.768085i \(-0.278790\pi\)
0.640348 + 0.768085i \(0.278790\pi\)
\(864\) 0 0
\(865\) −18.1168 −0.615991
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.82473 + 4.89258i −0.0958225 + 0.165970i
\(870\) 0 0
\(871\) 5.85300 + 10.1377i 0.198321 + 0.343502i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −10.8817 18.8477i −0.367870 0.637170i
\(876\) 0 0
\(877\) 19.1753 33.2125i 0.647503 1.12151i −0.336215 0.941785i \(-0.609147\pi\)
0.983717 0.179722i \(-0.0575199\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.2337 1.01860 0.509299 0.860589i \(-0.329905\pi\)
0.509299 + 0.860589i \(0.329905\pi\)
\(882\) 0 0
\(883\) 32.0446 1.07839 0.539193 0.842182i \(-0.318729\pi\)
0.539193 + 0.842182i \(0.318729\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 10.8817 18.8477i 0.365373 0.632845i −0.623463 0.781853i \(-0.714275\pi\)
0.988836 + 0.149008i \(0.0476080\pi\)
\(888\) 0 0
\(889\) 26.2337 + 45.4381i 0.879850 + 1.52394i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −13.1168 22.7190i −0.438938 0.760264i
\(894\) 0 0
\(895\) −31.7187 + 54.9384i −1.06024 + 1.83639i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −34.6708 −1.15633
\(900\) 0 0
\(901\) −9.48913 −0.316129
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −44.2337 + 76.6150i −1.47038 + 2.54677i
\(906\) 0 0
\(907\) 17.1724 + 29.7435i 0.570201 + 0.987618i 0.996545 + 0.0830565i \(0.0264682\pi\)
−0.426343 + 0.904561i \(0.640198\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 11.4312 + 19.7995i 0.378734 + 0.655987i 0.990878 0.134759i \(-0.0430262\pi\)
−0.612144 + 0.790746i \(0.709693\pi\)
\(912\) 0 0
\(913\) 1.29211 2.23800i 0.0427626 0.0740670i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 30.3505 1.00226
\(918\) 0 0
\(919\) 26.4663 0.873044 0.436522 0.899694i \(-0.356210\pi\)
0.436522 + 0.899694i \(0.356210\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −6.45364 + 11.1780i −0.212424 + 0.367929i
\(924\) 0 0
\(925\) 12.7446 + 22.0742i 0.419039 + 0.725796i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 15.3139 + 26.5244i 0.502431 + 0.870237i 0.999996 + 0.00280985i \(0.000894404\pi\)
−0.497565 + 0.867427i \(0.665772\pi\)
\(930\) 0 0
\(931\) −42.1627 + 73.0280i −1.38183 + 2.39340i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.00314 0.229027
\(936\) 0 0
\(937\) −22.2337 −0.726343 −0.363171 0.931722i \(-0.618306\pi\)
−0.363171 + 0.931722i \(0.618306\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −8.54755 + 14.8048i −0.278642 + 0.482622i −0.971048 0.238886i \(-0.923218\pi\)
0.692405 + 0.721509i \(0.256551\pi\)
\(942\) 0 0
\(943\) 2.35143 + 4.07279i 0.0765730 + 0.132628i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.51437 + 4.35502i 0.0817062 + 0.141519i 0.903983 0.427569i \(-0.140630\pi\)
−0.822277 + 0.569088i \(0.807296\pi\)
\(948\) 0 0
\(949\) −7.11684 + 12.3267i −0.231023 + 0.400143i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −8.60597 −0.278775 −0.139387 0.990238i \(-0.544513\pi\)
−0.139387 + 0.990238i \(0.544513\pi\)
\(954\) 0 0
\(955\) 59.3863 1.92170
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −24.6644 + 42.7200i −0.796456 + 1.37950i
\(960\) 0 0
\(961\) −5.32473 9.22271i −0.171766 0.297507i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.68614 + 2.92048i 0.0542788 + 0.0940136i
\(966\) 0 0
\(967\) 24.6644 42.7200i 0.793154 1.37378i −0.130850 0.991402i \(-0.541771\pi\)
0.924005 0.382381i \(-0.124896\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 57.5333 1.84633 0.923165 0.384404i \(-0.125593\pi\)
0.923165 + 0.384404i \(0.125593\pi\)
\(972\) 0 0
\(973\) −92.5842 −2.96811
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 22.6168 39.1735i 0.723577 1.25327i −0.235980 0.971758i \(-0.575830\pi\)
0.959557 0.281514i \(-0.0908366\pi\)
\(978\) 0 0
\(979\) −5.57825 9.66181i −0.178282 0.308793i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −29.6932 51.4301i −0.947065 1.64036i −0.751563 0.659661i \(-0.770700\pi\)
−0.195502 0.980703i \(-0.562633\pi\)
\(984\) 0 0
\(985\) −18.1168 + 31.3793i −0.577251 + 0.999827i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.11684 −0.130908
\(990\) 0 0
\(991\) −7.00314 −0.222462 −0.111231 0.993795i \(-0.535479\pi\)
−0.111231 + 0.993795i \(0.535479\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −28.7666 + 49.8253i −0.911963 + 1.57957i
\(996\) 0 0
\(997\) −5.17527 8.96382i −0.163902 0.283887i 0.772363 0.635182i \(-0.219075\pi\)
−0.936265 + 0.351295i \(0.885742\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1728.2.i.n.577.4 8
3.2 odd 2 576.2.i.n.193.2 8
4.3 odd 2 inner 1728.2.i.n.577.3 8
8.3 odd 2 864.2.i.f.577.1 8
8.5 even 2 864.2.i.f.577.2 8
9.2 odd 6 576.2.i.n.385.2 8
9.4 even 3 5184.2.a.cc.1.1 4
9.5 odd 6 5184.2.a.cf.1.3 4
9.7 even 3 inner 1728.2.i.n.1153.4 8
12.11 even 2 576.2.i.n.193.3 8
24.5 odd 2 288.2.i.f.193.3 yes 8
24.11 even 2 288.2.i.f.193.2 yes 8
36.7 odd 6 inner 1728.2.i.n.1153.3 8
36.11 even 6 576.2.i.n.385.3 8
36.23 even 6 5184.2.a.cf.1.4 4
36.31 odd 6 5184.2.a.cc.1.2 4
72.5 odd 6 2592.2.a.u.1.1 4
72.11 even 6 288.2.i.f.97.2 8
72.13 even 6 2592.2.a.x.1.3 4
72.29 odd 6 288.2.i.f.97.3 yes 8
72.43 odd 6 864.2.i.f.289.1 8
72.59 even 6 2592.2.a.u.1.2 4
72.61 even 6 864.2.i.f.289.2 8
72.67 odd 6 2592.2.a.x.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.f.97.2 8 72.11 even 6
288.2.i.f.97.3 yes 8 72.29 odd 6
288.2.i.f.193.2 yes 8 24.11 even 2
288.2.i.f.193.3 yes 8 24.5 odd 2
576.2.i.n.193.2 8 3.2 odd 2
576.2.i.n.193.3 8 12.11 even 2
576.2.i.n.385.2 8 9.2 odd 6
576.2.i.n.385.3 8 36.11 even 6
864.2.i.f.289.1 8 72.43 odd 6
864.2.i.f.289.2 8 72.61 even 6
864.2.i.f.577.1 8 8.3 odd 2
864.2.i.f.577.2 8 8.5 even 2
1728.2.i.n.577.3 8 4.3 odd 2 inner
1728.2.i.n.577.4 8 1.1 even 1 trivial
1728.2.i.n.1153.3 8 36.7 odd 6 inner
1728.2.i.n.1153.4 8 9.7 even 3 inner
2592.2.a.u.1.1 4 72.5 odd 6
2592.2.a.u.1.2 4 72.59 even 6
2592.2.a.x.1.3 4 72.13 even 6
2592.2.a.x.1.4 4 72.67 odd 6
5184.2.a.cc.1.1 4 9.4 even 3
5184.2.a.cc.1.2 4 36.31 odd 6
5184.2.a.cf.1.3 4 9.5 odd 6
5184.2.a.cf.1.4 4 36.23 even 6